\ln\frac{n+k}{n+1}=\int_{1+\frac 1n}^{1+\frac kn}\frac{dx}x<\frac 1n\left(\frac 1{1+\frac 1n}+\cdots+\frac 1{1+\frac 1{n+k}}\right)<\int_1^{1+\frac{k-1}n}\frac{dx}x=\ln\frac{m+k-1}m